Unoriented bordism
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Contents |
1 Introduction
We denote the non-oriented bordism groups by . The sum of these groups
are a ring under cartesian products of the manifolds. Thom [Thom1954] has shown that this ring is a polynomial ring over in variables for and he has shown that for even one can take for . Dold [Dold1956] has constructed manifolds for with odd.
2 Construction and examples
Dold constructs certain bundles over with fibre denoted by
Using the results by Thom [Thom1954] Dold shows that these manifolds give ring generators of .
Theorem (Dold) [Dold1956] 2.1. For even set and for set . Then for
are polynomial generators of olver :
3 Invariants
To prove the Theorem Dold has to compute the characteristic numbers which according to Thom's theorem determine the bordism class. As a first step Dold computes the cohomology ring with -coeffcients. The fibre bundle has a section and we consider the cohomology classes (always with -coefficients)
where is a generator of , and
which is characterized by the property that the restriction to a fibre is non-trivial and .
Theorem [Dold1956] 3.1. The classes and generate with only relation
and
The Steenrod squares act by
and all other Squares act trivially on and . On the decomposable classes the action is given by the Cartan formula.
The total Stiefel-Whitney class of the tangent bundle is
4 Classification/Characterization (if available)
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5 Further discussion
For odd the manifolds are orientable and thus after choosing an orientation give an element in the oriented bordism group . Since admits an obvious orientation reversing diffeomorphism, these elements are -torsion. Thus we obtain a subring in isomorphic to . For more information about see the page on oriented bordism.
6 References
- [Dold1956] A. Dold, Erzeugende der Thomschen Algebra , Math. Z. 65 (1956), 25–35. MR0079269 (18,60c) Zbl 0071.17601
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
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